do with the self. Conversely, for socioemotional leadership, q is greater than 1: The self is a better informant concerning socioemotional leadership than are the other group members.
In addition, we examined whether the self-mean and dyadic means were different. As can be seen in Table 4, they were not. Although mean differences were found, they were rather small. These results are consistent with the results of Anderson et al. (2006), who found little or no self-enhancement for perceptions of status.
We also examined whether relationship variance differed for the percep-tion of self and others. We did find a large difference for socioemopercep-tional leadership. Contrary to our expectations, less relationship variance was found in self-ratings than in the ratings of others. Another explanation for the task and socioemotional differences in self-enhancement is that socioemotional leadership is more closely linked to group belongingness than to more observable task activities. In essence, members refrain from making judgments that decrease social acceptance so as to maintain their belongingness in a group (Anderson et al., 2006).
Generally, for both types of leadership, the parameter estimates are not all dramatically different in the two types of ratings, self and other. There is little or no self-enhancement or effacement, and both k and q are close to 1.
Although some differences in relationship variance are apparent, they arise only with socioemotional leadership. The bottom line is that it would not be especially problematic to include self-ratings in the measurement of the target and perceiver effects.
ESTIMATION OF SRM USING SPECIALIZED
with partner j in group k is
Xijk ¼mkþaikþbjkþgijk
where Xijkis the score for person i rating (or behaving with) person j, mkthe group mean, aikperson i’s actor effect, bjkperson j’s partner effect, and gijk
the relationship effect. The terms m, a, b, and g are random variables and the following variances are parameters of the model: s2m; s2a; s2b; and s2g.
The SRM also specifies two different correlations between the SRM components of a variable, both of which can be viewed as reciprocity correlations:
At the individual level, a person’s actor effect can be correlated with that person’s partner effect and can be denoted as sab, generalized reciprocity.
At the dyadic level, two members’ relationship effects can be correlated and can be denoted as sgg, dyadic reciprocity.
The final parameter is the mean of mkor m, which is an overall intercept or
‘‘grand mean.’’ Thus, there are seven univariate SRM parameters: one mean, four variances, and two covariances.
ANOVA Estimation
Prior to the current work, almost all published papers used the method of moments – sometimes called the ANOVA (analysis of variance) method – to estimate these variances and covariances. This method is described in some detail in Kenny, Kashy, and Cook (2006). Basically, a term is computed, such as an estimate of the target effect; its variance is computed; and then the expected value of that variance is determined. This process is then repeated for the other SRM components. What results is a series of simultaneous equations, such that linear algebra can be used to solve for the unknown SRM variances and covariances from the variances and covariances of the estimated components. A computer program for the estimation of these components called SOREMO (SOcial RElations MOdel;
Kenny, 1998) has been developed to estimate the SRM parameters from round-robin designs.
In our example, we use data gathered byLord et al. (1980). Their study, which involved a total of 24 four-person groups, measured how much each person in the group stated the other member contributed to the group, using a rating scale from 1 to 6. We denote this variable as LEAD. The data
structure is a round-robin design, which has an n n structure in which the diagonal (self-ratings) is missing.
Table 5 presents a summary of the results from the computer program SOREMO. We now consider how conventional software can estimate the SRM variances and covariances.Table 1provides the SOREMO estimates of the seven parameters, which include a negative estimate of the group variance.
Conventional Multi-Level Modeling: SAS and SPSS
Increasingly, multi-level models can estimate models with cross-classified or crossed variables. In these models, the actor–partner covariance is assumed to be zero, which is a major limitation of this method. We describe this approach in three steps, and describe how both SAS and SPSS can be used to carry out the model’s calculations. Presumably, the program HLM could be used for this purpose as well.
Step 1: Data Organization and Preparation
Create a data set in which each record is the response of one person about another person in a dyad on all variables (e.g., person A’s rating of person B on leadership, talkativeness, and intelligence). For a one round-robin of 5 persons, there would be 20 records, assuming that self-ratings are not included. Make sure the following variables are included in each record:
A unique actor number: For example, for group 1, the actor numbers would range from 1 to 5, and for group 2, the actor numbers would range from 6 to 10.
Table 5. Summary of Results Using Different Programs.
Term Symbol SOREMO SAS Ia SPSSa SAS IIb SEM
Mean m 3.868 3.868 3.868 3.868 3.868
Actor variance s2a 0.233 0.198 0.198 0.198 0.233
Partner variance s2b 0.240 0.192 0.192 0.204 0.240
Group variance s2m 0.091 0.000 0.000 0.000 0.094
Actor–partner covariance sab 0.059 0.000c 0.000c 0.024 0.059
Error variance s2g 0.222 0.237 0.237 0.230 0.222
Error covariance sgg 0.014 0.032 0.032 0.022 0.014
aActor–partner covariance is assumed to be zero.
bDummy variables with equality constraints.
cFixed to zero.
A unique partner number.
A unique dyad number: For a five-person group, there are 10 dyads.
A unique group number.
Step 2: Syntax
We first present the syntax for SAS, followed by the syntax for SPSS.
Note that the actor–partner covariance is not modeled and that LEAD is the outcome of interest.
The syntax for SAS is as follows:
PROC MIXED COVTEST;
CLASS ACTOR PARTNER DYAD GROUP;
MODEL LEAD ¼ /S DDFM ¼ SATTERTH NOTEST;
RANDOM INTERCEPT/TYPE ¼ VC SUB ¼ ACTOR;
RANDOM INTERCEPT/TYPE ¼ VC SUB ¼ PARTNER;
RANDOM INTERCEPT/TYPE ¼ VC SUB ¼ GROUP;
REPEATED/TYPE ¼ CS SUB ¼ DYAD;
The syntax for SPSS is as follows:
MIXED
LEAD BY GROUP /FIXED ¼
/PRINT ¼ SOLUTION TESTCOV
/RANDOM INTERCEPT|SUBJECT(GROUP) COVTYPE(VC) /RANDOM INTERCEPT|SUBJECT(ACTOR) COVTYPE(VC) /RANDOM INTERCEPT|SUBJECT(PARTNER) COVTYPE(VC) /RANDOM INTERCEPT|SUBJECT(DYAD) COVTYPE(VC).
With SPSS, the REPEATED statement cannot be used for the dyad, so one must presume that the dyadic covariance is positive. Also, for SPSS, error variance equals the dyad variance plus the error variance, and the dyadic correlation equals the dyad variance divided by the sum of the dyad variance plus the error variance.
As can be seen inTable 5, the SAS and the SPSS programs yield the same results. These outcomes would have been different if the reciprocity covariance were negative. In that case, it would be estimated as zero by SPSS and properly estimated by SAS. Note also that the estimates are different from SOREMO. The major reason for this difference is the assumption of the zero actor–partner covariance. Because that covariance is small, the differences are small. As discussed earlier in this chapter, in the
study of leadership, the actor–partner covariance is typically small, so this method can be used fruitfully.
SAS with Dummy Variables
An approach to estimating the full SRM was initially proposed bySnijders and Kenny (1999) and illustrated using the program MLwiN. Here, we apply their approach but use the package SAS. Currently, this approach cannot be applied using SPSS and HLM. We warn the reader that this approach is very complicated.
With this approach, 2n dummy random variables are created and constraints are placed on the variance–covariance matrix of those dummy variables. We describe their approach in three steps. We used this approach to analyze gender effects in theLord et al. (1978)data earlier in this chapter.
This method is most appropriate when explicit predictors (i.e., fixed effects) of the SRM components are available.
Step 1: Data Organization and Preparation
Create a data set where each record refers to a data point or n(n1) data points for each group, where n is the group size (assuming no self-data). For each observation, we have a variable that designates which group the person is in, which dummy is used, and which observation is relevant. We create two sets of dummy variables:
A(1) through A(n), where n is the largest group size. For a dummy variable A(i), if the actor is person i, the dummy equals 1; otherwise, the dummy equals 0.
P(1) through P(n), where n is the largest group size. For a dummy variable P(i), if the partner is person i, the dummy equals 1; otherwise, the dummy equals 0.
The following SAS code might be used to create the dummy variables for a four-person round-robin in which there is a variable for actor and partner that goes from 1 to 4:
A1 ¼ 0; A2 ¼ 0; A3 ¼ 0; A4 ¼ 0;
IF ACT ¼ 1 THEN A1 ¼ 1;
IF ACT ¼ 2 THEN A2 ¼ 1;
IF ACT ¼ 3 THEN A3 ¼ 1;
IF ACT ¼ 4 THEN A4 ¼ 1;
P1 ¼ 0; P2 ¼ 0; P3 ¼ 0; P4 ¼ 0;
IF PART ¼ 1 THEN P1 ¼ 1;
IF PART ¼ 2 THEN P2 ¼ 1;
IF PART ¼ 3 THEN P3 ¼ 1;
IF PART ¼ 4 THEN P4 ¼ 1;
Step 2: Force Constraints
This step is the least intuitive, but the most important part of the analysis.
We need to place constraints on the variance–covariance matrix of the dummy variables. For instance, the covariances of the actor’s effects are set to zero. So far as we know, SPSS and HLM (but not MLwiN) do not allow for such constraints.
To do so, we create a data file, in this case called G, to set the n actor variances (parameter 1) equal, the n partner variances (parameter 2) equal, and the n actor–partner covariances (parameter 3) equal. The file for a four-person group has the following structure:
DATA G;
INPUT PARM ROW COL VALUE;
DATALINES;
1 1 1 1 1 2 2 1 1 3 3 1 1 4 4 1 2 5 5 1 2 6 6 1 2 7 7 1 2 8 8 1 3 1 5 1 3 2 6 1 3 3 7 1 3 4 8 1 4 9 9 1;
The four values in each record in the data file are input in the following order:
Parameter number (where 1 refers to actor variance, 2 refers to partner variance, 3 refers to actor–partner covariance, and 4 refers to group variance).
Row of the variance–covariance matrix.
Column of the matrix.
Value in the matrix (almost always 1).
For instance, the first line refers to actor variance in row 1 and column 1 and that value is set to 1.
In summary, the first four lines set the actor variances equal, the next four lines set the partner variances equal, the next four lines set the actor–partner covariances equal, and the last line refers to the group variance. If an element is not referenced, then it is fixed to zero.
Step 3: Run the Multi-Level Model
Following is the SAS code for PROC MIXED:
PROC MIXED COVTEST;
CLASS DYAD GROUP;
MODEL LEAD ¼ /S DDFM ¼ SATTERTH NOTEST;
RANDOM A1 A2 A3 A4 P1 P2 P3 P4 INTERCEPT /G SUB ¼ GROUP TYPE ¼ LIN(4) LDATA ¼ G;
REPEATED/TYPE ¼ CS SUB ¼ DYAD(GROUP);
Note that the ‘‘ldata ¼ g’’ statement in the RANDOM statement sets the equality constraints. There are nine (2nþ1) terms in the RANDOM statement, A1 through INTERCEPT, which are ordered as in G (actor first, then partner, and finally group). If the model included predictor variables (e.g., gender), they would be included in the MODEL record after the equality sign, but before the slash.
As seen inTable 5, the results obtained with SAS are similar to those yielded by SOREMO. We did not include the analysis using MLwiN, but the results are essentially the same estimates as those produced by SAS, even though the two programs use somewhat different estimation methods. This method is currently complex, but we would expect that it will eventually become simpler to use and be adapted to other packages (e.g., SPSS and HLM).
Structural Equation Modeling
The method of SEM is a generalization of the method developed byOlsen and Kenny (2006)for dyadic analysis. When we analyzed the relationship between self- and other perception earlier in this chapter, we used an extension of the SEM approach. This method is not especially difficult, but it is messy and awkward.
Step 1: Data Preparation
Unlike in the other methods, the group is the unit of analysis with the SEM approach. If n members are in the largest group size, there would be n(n1) scores read per group. For n ¼ 4, the variables would be X12, X13, X14, X21, X23, X24, X31, X32, X34, X41, X42, and X43. Self-measures and other predictors could be included. The order of the variables does not matter, and scores can be missing. Normally, there must be at least two more groups (i.e., ‘‘individuals’’) than the number of variables to use SEM.
However, Amos can accommodate in the case in which there are fewer groups than variables if one tells the program to allow for nonpositive definite input matrices. So far as we know, this option is not currently available in other SEM computer programs.
Step 2: Latent Variables
There would be n actor factors, n partner factors, and one additional latent variable for the group. A given measure loads on its actor factor, a partner factor, and the general mean factor. For instance, the measure X32 loads on the actor factor for person 3, the partner factor for person 2, and the mean factor. All factor loadings are fixed to 1.
Parallel actor and partner effects would be correlated. Thus, the actor factor for person 1 would be correlated with the partner factor for person 1.
All other factor correlations are fixed to zero. Additionally, there would be correlations between pairs of errors for the two members of the dyad (e.g., the errors of X12 and X21).
Step 3: Equality Constraints
To achieve an identified model, many equality constraints must be applied.
The n(n1) means would be set equal, as well as the n actor variances, the n partner variances, the n(n1) relationship variances, the n actor–partner covariances, and the n(n1)/2 error covariances. Because a large number of equality constraints are made and many parameters are fixed to 1 (i.e., all factor loadings), the total number of parameters in the model is only seven.
Thus, if n is 4, there are 78 elements in the moment matrix, and 71 degrees of freedom in the model.
Step 4: Model Testing
The fit of the model does not matter, because the fit varies depending who is assigned to persons 1, 2, 3, and so on. The model is treated as an I-SAT (i.e., interchangeable saturated) model, as described byOlsen and Kenny (2006).
Nevertheless, specialized models (e.g., setting actor and partner variances
equal) can be estimated, and the changes in fit in these restricted models from the saturated model can be ascertained. Also note that the estimates are maximum likelihood estimates, rather than restricted maximum likelihood estimates like those obtained in the multi-level modeling program.
We used the program AmosBasic to estimate the model. Using AmosGraphics worked, but the model included so many variables and constraints that it was difficult to implement and verify its correctness.
(This is why we described this approach as ‘‘messy.’’) We suggest always outputting the ‘‘implied moments,’’ to determine whether the constraints were successfully implemented.
Comparison of Different Methods
Each of the various methods has its own advantages and disadvantages.
We believe that the SAS dummy variable estimates would be the same as those produced with SOREMO, which uses the ANOVA method when there are equal group sizes, no missing data, and all variances greater than or equal to zero. SEM results are slightly biased because the program uses a maximum likelihood estimation. Note that SOREMO and SEM do allow for negative variances (although some SEM programs can prevent negative estimates of variance). The multi-level program MLwiN offers the option of allowing for negative variances. In the case of our example, we obtain the value of 0.091 for the group variance with this approach.
Using conventional software offers several advantages over using SOREMO. First, conventional software can handle the situation in which there are missing data. By contrast, SOREMO presumes that there are no missing data; if there are, either the entire group must be removed or some method must be used to impute missing values. Second, SOREMO requires that the group contain a minimum of four people. With these alternative methods, smaller-sized groups can be retained. Third, when group sizes are unequal, the results from different groups are optimally weighted. Fourth, one can estimate specialized models, such as a model that sets group variance to zero, a model that sets the actor–partner and relationship covariances to zero, or a model that sets the actor and partner variances equal. SOREMO estimates only the ‘‘saturated model’’ – that is, the model with all parameters.
For instance, using SAS with dummy variables and setting the group variance to zero with our example yields actor variance of 0.1989, partner variance of 0.2056, actor–partner covariance of 0.04404, dyadic covariance of 0.03828, relationship variance of 0.2098, and intercept of 3.8640.
The one major advantage of SOREMO is that it can estimate, in a single run, all the variances and correlations for a large number of variables.
Perhaps, SOREMO might be used for initial analyses of the variables, and these other programs used once more specific models are available.
The discussion here has considered only univariate models. Elsewhere, we have used the dummy variable approach with SAS to estimate a bivariate model (Kenny, West, & Cillessen, 2007). In this case, we constructed two additional dummy variables for the means of each variable and fixed the error variance to a very small value, essentially zero. We have also used the SEM approach to estimate a path model in which the actor and partner effects caused self-ratings (Kenny & West, 2008), a method applied earlier in this chapter.
CONCLUSION
Although this chapter has highlighted many interesting and theoretically important uses of the SRM in the study of leadership, it has not discussed all of the possibilities. Here, we briefly review other types of analyses that could be performed using this model.
Further Extensions
Two major topics were omitted from this chapter: group-level outcomes (e.g., group productivity) and data gathered from persons who are members of multiple groups or persons who are not members of the group (e.g., observers). We briefly consider each of these topics here.
Some of the major outcomes of analyses in the study of leadership in natural settings are at the level of the group: group productivity, cohesiveness, and type of team (Greguras, Robie, & Born, 2001). Group-level predictor variables are not problematic, they can just be added as predictors in the model. For instance, we considered a group-level predictor of gender composition when we discussed gender differences. Group-level outcomes, however, do present challenges.
To study group outcomes, we can follow the pioneering work ofDabbs and Ruback (1987). They performed a separate SRM analysis for each group and correlated SRM variances and covariances with group-level outcomes. For instance, we could ask whether agreement about the target
effect (i.e., the proportion of total variance that is due to the target) predicted group productivity.
Sometimes, the same person may be a member of more than one group.
One key question is whether a component varies across different group compositions and tasks. A rotation design (Kenny & Hallmark, 1992) places the same person in multiple groups with different persons; the analysis then considers whether the person is seen the same way in all of those groups.
Using a rotation design, which involves a series of carefully constructed round-robin designs, Zaccaro et al. (1991) examined the extent to which the target effect correlated across four different tasks. In their study, the correlation of the target effect of leadership across four highly hetero-geneous tasks was 0.59. More recently,Foti and Hauenstein (2007) used a rotation design to study leadership in a military context. They found that the correlation of the target effect of leadership across heterogeneous tasks was 0.44. It is also possible to examine the stability of perceiver and even relationship effects (If Alex is in a group with Bob, is it more likely that Alex will be a leader?). For more information about the rotation design, consult Kenny and Hallmark (1992).
In our earlier example, we included data from only the members of the group. Often, data are available from other persons, besides those who are members of the group. For instance, others might observe or supervise the group. If multiple people observe each group member (as in studies of 360-degree feedback;Yammarino & Atwater, 1997), the SRM design called block design is applicable (Kenny, 1994). With this design, one set of persons constitutes the perceivers and the other set forms the targets.
In addition, sometimes the group interacts with another group. If there are ratings of persons in the other group or out-group judgments, the study is said to have a block round-robin design. A very interesting question is whether someone who is seen as a leader of his or her own group is also seen as a leader by members of the other group.Boldry and Kashy (1999) have tested this interesting hypothesis and have found agreement in the perceptions of leadership across groups (r ¼ 0.52).
Correlations of Components
This chapter began with a traditional partitioning of variance into SRM components. If we are to understand the theoretical meaning of those components, we need to determine the correlations between variables.
Earlier in this chapter, we illustrated the correlations between two random